Algebraic number

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, $(1+{\sqrt {5}})/2$ , is an algebraic number, because it is a root of the polynomial $x 2 - x - 1$ . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number $1+i$ is algebraic because it is a root of $x 4 + 4$ .

All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as $π$ and $e$ , are called transcendental numbers.

The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.

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